The maximum and minimum values of
`-4le5cos theta+3cos(theta+(pi)/(3))+3le10` are respectively

Options

A. `4 and -4`
B. `10 and -4`
C. `10 and -10`
D. `6and -4`

Explanation

To find the maximum and minimum values of the expression \(-4 \leq 5\cos(\theta) + 3\cos\left(\theta + \frac{\pi}{3}\right) + 3 \leq 10\), we will follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ 5\cos(\theta) + 3\cos\left(\theta + \frac{\pi}{3}\right) + 3 \]
Using the cosine addition formula, we can rewrite \(\cos\left(\theta + \frac{\pi}{3}\right)\):
\[ \cos\left(\theta + \frac{\pi}{3}\right) = \cos(\theta)\cos\left(\frac{\pi}{3}\right) - \sin(\theta)\sin\left(\frac{\pi}{3}\right) \]
Since \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\) and \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\), we have:
\[ \cos\left(\theta + \frac{\pi}{3}\right) = \frac{1}{2}\cos(\theta) - \frac{\sqrt{3}}{2}\sin(\theta) \]

### Step 2: Substitute Back into the Expression
Substituting this back into the expression gives:
\[ 5\cos(\theta) + 3\left(\frac{1}{2}\cos(\theta) - \frac{\sqrt{3}}{2}\sin(\theta)\right) + 3 \]
This simplifies to:
\[ 5\cos(\theta) + \frac{3}{2}\cos(\theta) - \frac{3\sqrt{3}}{2}\sin(\theta) + 3 \]
Combining the cosine terms:
\[ \left(5 + \frac{3}{2}\right)\cos(\theta) - \frac{3\sqrt{3}}{2}\sin(\theta) + 3 = \frac{13}{2}\cos(\theta) - \frac{3\sqrt{3}}{2}\sin(\theta) + 3 \]

### Step 3: Find the Maximum and Minimum Values
Now, we need to find the maximum and minimum values of:
\[ \frac{13}{2}\cos(\theta) - \frac{3\sqrt{3}}{2}\sin(\theta) \]
This can be expressed in the form \(R\cos(\theta + \phi)\), where:
\[ R = \sqrt{\left(\frac{13}{2}\right)^2 + \left(-\frac{3\sqrt{3}}{2}\right)^2} \]
Calculating \(R\):
\[ R = \sqrt{\frac{169}{4} + \frac{27}{4}} = \sqrt{\frac{196}{4}} = \sqrt{49} = 7 \]
Thus, the maximum value of \(\frac{13}{2}\cos(\theta) - \frac{3\sqrt{3}}{2}\sin(\theta)\) is \(7\) and the minimum value is \(-7\).

### Step 4: Adjust for the Constant
Now, we need to add \(3\) to both the maximum and minimum values:
- Maximum: \(7 + 3 = 10\)
- Minimum: \(-7 + 3 = -4\)

### Final Result
Thus, the maximum and minimum values of the original expression are:
\[ \text{Maximum value} = 10, \quad \text{Minimum value} = -4 \]

### Summary
The maximum and minimum values of the expression \(-4 \leq 5\cos(\theta) + 3\cos\left(\theta + \frac{\pi}{3}\right) + 3 \leq 10\) are \(10\) and \(-4\), respectively.